RANK 2 ACM BUNDLES WITH TRIVIAL DETERMINANT ON FANO THREEFOLDS OF GENUS 7

RANK 2 ACM BUNDLES WITH TRIVIAL DETERMINANTON FANO THREEFOLDS OF GENUS 7

MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

Abstract. Given a smooth prime Fano threefold X of genus 7, weprove that the subset M`f

X (2, 0, 4) of vector bundles in the moduli spaceMX(2, 0, 4) of rank 2 semistable sheaves on X with c1 = 0 and c2 =4 is an open dense subset of the Brill-Noether locus W 1

2,4 of rank 2stable sheaves with degree 4 with 2 sections, defined on the homologicallyprojectively dual curve Γ. This shows that M`f

X (2, 0, 4) is a smoothirreducible 5-fold.

1. Introduction

In this paper we investigate the moduli spaces of rank 2 stable vectorbundles on a smooth prime Fano threefold, carrying on the work taken upin [BF07] and [BF08a].

A smooth complex projective threefold X is called Fano if its anticanon-ical divisor −KX is ample. A Fano threefold X is prime if its Picard groupis generated by the class of KX . These varieties are classified up to de-formation, see for instance [IP99, Chapter IV]. The number of deformationclasses is 10, and they are characterized by the genus, which is the integer gsuch that deg(X) = −K3

X = 2 g − 2. Recall that the genus of a prime Fanothreefold take values in {2, . . . , 10, 12}.

Let now X be a smooth prime Fano threefold. We are interested in theMaruyama moduli scheme MX(2, c1, c2) of semistable sheaves F on X ofrank 2 with Chern classes c1, c2, and with c3 = 0. We will be particularlyinterested in the subset of MX(2, c1, c2) consisting of arithmetically Cohen-Macaulay (ACM) bundles, i.e. satisfying Hk(X, F (t)) = 0 for all t and fork = 1, 2. Since the rank of F is 2, we can assume c1 ∈ {0, 1}. We denote byM`f

X (2, c1, c2) the subset of locally free sheaves in MX(2, c1, c2).The geometry of these moduli spaces has been mostly studied for c1 =

1, and many results in the literature concern specific values of c2. Forinstance, if one asks whether the moduli space MX(2, 1, c2) is smooth andirreducible, then the answer is known (most frequently in the affirmativesense) only for low values of c2. Low here means close to mg = d(g + 2)/2e,indeed MX(2, 1, c2) is empty for c2 < mg. For higher values of c2, thespace MX(2, 1, c2) is known to contain a reduced component of dimension

2000 Mathematics Subject Classification. Primary 14J60. Secondary 14H30, 14F05,14D20.

Key words and phrases. Prime Fano threefolds. Moduli space of vector bundles.Semiorthogonal decomposition. Brill-Noether theory for vector bundles on curves.

Both authors were partially supported by Italian MIUR funds. The first author wouldlike to thank the University of Pau, where this paper has been completed, for its hospitality.

1

2 MARIA CHIARA BRAMBILLA AND DANIELE FAENZI

2c2 − g − 2. We refer for more details to the papers [IM00b] (for genus 3),[IM04a], [IM07], [BF07] (for genus 7), [IM07a], [IM00a] (for genus 8), [IR05][BF08b] (for genus 9), [AF06] (for genus 12), [BF08a] (for all genera). Westress that MX(2, 1, c2) contains a component whose general element is astable ACM bundle if and only if mg ≤ c2 ≤ g + 3.

Much less has been said on the case of trivial determinant, i.e. whenc1 = 0. One sees easily that M`f

X (2, 0, c2) is empty unless c2 is even andgreater than 2, so the first case to study is M`f

X (2, 0, 4). On the other hand,a sheaf in M`f

X (2, 0, c2) can be an ACM bundle if and only if c2 = 4.The study of the space M`f

X (2, 0, 4) was first taken up by Iliev and Marku-shevich in [IM00b, arXiv version] for genus 3. In this case they proved thatM`f

X (2, 0, 4) has two irreducible components. Assume now g ≥ 4. In viewof [BF08a], we know that M`f

X (2, 0, 4) contains a component of dimension 5,for all smooth prime Fano threefolds of genus g. Again a general element ofthis component is a stable ACM bundle.

In this paper we study the space M`fX (2, 0, 4), where X is a smooth prime

Fano threefold of genus 7. We use the semiorthogonal decomposition of thebounded derived category Db(X) obtained by Kuznetsov in [Kuz05]. Moreprecisely, we consider the homologically projectively dual curve Γ in thesense of [Kuz06]. Recall that Γ is smooth non-hyperelliptic curve of genus7, and that there is a natural integral functor Φ! : Db(X) → Db(Γ).

Here we first prove that, given any sheaf F in M`fX (2, 0, 4), the sheaf F (1)

is mapped by Φ! to a complex concentrated in degree −1, so the shiftedcomplex F = Φ!(F (1))[−1] is in fact a locally free sheaf on Γ. The bundleF then turns out to belong to the Brill-Noether variety W 1

2,4(Γ) of rank 2stable bundles on Γ with degree 4 with at least 2 independent global sections.Finally, we remark that any element in M`f

X (2, 0, 4) is an ACM bundle. Thisleads to our main result:

Theorem. Let X be a smooth prime Fano threefold of genus 7 and letM`f

X (2, 0, 4) be the subset of locally free sheaves in MX(2, 0, 4). Then themap ϕ defined by:

M`fX (2, 0, 4) → W 1

2,4(Γ)

F 7→ Φ!(F (1))[−1]

is an open immersion. In particular, the moduli space M`fX (2, 0, 4) is a

smooth irreducible variety of dimension 5. Any element of this space isa stable ACM bundle.

Here is the structure of our paper. In the next section we recall a fewpreliminary notions, while in section 3 we prove some preparatory vanishingresults. Section 4 is devoted to the proof of our main theorem. Finally,in section 5 we show that, if S is a general hyperplane section surface ofX, then the space M`f

X (2, 0, 4) is embedded in M`fS (2, 0, 4) as a lagrangian

subvariety.

BUNDLES ON FANO THREEFOLDS OF GENUS 7 3

2. Preliminaries

Given a smooth complex projective n-dimensional polarized variety(X, HX) and a sheaf F on X, we write F (t) for F ⊗OX(tHX). Givena pair of sheaves (F,E) on X, we will write extk

X(F,E) for the dimen-sion of the Cech cohomology group Extk

X(F,E), and similarly hk(X, F ) =dim Hk(X, F ). The Euler characteristic of (F,E) is defined as χ(F,E) =∑

k(−1)k extkX(F,E) and χ(F ) is defined as χ(OX , F ). We denote by p(F, t)

the Hilbert polynomial χ(F (t)) of the sheaf F . The degree deg(L) of a di-visor class L is defined as the degree of L ·Hn−1

X . The dualizing sheaf of Xis denoted by ωX .

If X is an smooth n-dimensional subvariety of Pm, whose coordinate ringis Cohen-Macaulay, then X is said to be arithmetically Cohen-Macaulay(ACM). A locally free sheaf F on an ACM variety X is said to be an ACMbundle if it has no intermediate cohomology, i.e. if Hk(X, F (t)) = 0 forall integer t and for any 0 < k < n. The corresponding module over thecoordinate ring of X is thus a maximal Cohen-Macaulay module.

Let us now recall a few well-known facts about semistable sheaves onprojective varieties. We refer to the book [HL97] for a more detailed accountof these notions. We recall that a torsionfree coherent sheaf F on X is(Gieseker) semistable if for any coherent subsheaf E, with 0 < rk(E) <rk(F ), one has p(E, t)/ rk(E) ≤ p(F, t)/ rk(F ) for t � 0. The sheaf F iscalled stable if the inequality above is always strict.

The slope of a sheaf F of positive rank is defined as µ(F ) =deg(c1(F ))/ rk(F ), where c1(F ) is the first Chern class of F . We recallthat a torsionfree coherent sheaf F is µ-semistable if for any coherent sub-sheaf E, with 0 < rk(E) < rk(F ), one has µ(E) < µ(F ). The sheaf F iscalled µ-stable if the above inequality is always strict. We recall that thediscriminant of a sheaf F is ∆(F ) = 2rc2(F ) − (r − 1)c1(F )2, where thek-th Chern class ck(F ) of F lies in Hk,k(X). Bogomolov’s inequality, see forinstance [HL97, Theorem 3.4.1], states that if F is also µ-semistable, thenwe have:

(2.1) ∆(F ) ·Hn−2X ≥ 0.

Recall that by Maruyama’s theorem, see [Mar80], if dim(X) = n ≥ 2 andF is a µ-semistable sheaf of rank r < n, then its restriction to a generalhypersurface of X is still µ-semistable.

We introduce here some notation concerning moduli spaces. We denoteby MX(r, c1, . . . , cn) the moduli space of S-equivalence classes of rank rtorsionfree semistable sheaves on X with Chern classes c1, . . . , cn. The Chernclass ck will be denoted by an integer as soon as Hk,k(X) has dimension 1.We will drop the last values of the classes ck when they are zero. We denoteby M`f

X (r, c1, . . . , cn) the subset of MX(r, c1, . . . , cn) given by locally freesheaves.

We will work with Brill-Noether varieties of vector bundles over a smoothprojective curve. We refer to [TiB91a] for some basic results. We willneed also some results from [TiB91b] and [Mer01]. By definition, the Brill-Noether variety W s

r,c(Γ) is the scheme parameterizing rank r µ-stable bundlesof degree c on Γ having at least s + 1 independent global sections. It has


expected dimension:

(2.2) ρ(r, c, s) = 6 r2 − (s + 1) (s + 1− c + 6 r) + 1.

Recall also that the Gieseker-Petri map associated to a sheaf F on Γ isdefined as the natural multiplication map:

(2.3) πF : H0(Γ,F)⊗H0(Γ,F∗⊗ωΓ) → H0(Γ,F ⊗F∗⊗ωΓ).

The map πF associated to a stable bundle F in MΓ(r, d) is injective ifand only if [F ] is a nonsingular point of a component of W s

r,d of dimensionρ(r, d, s). Its transpose has the form:

(2.4) π>F : Ext1Γ(F ,F) → H0(Γ,F)∗⊗H1(Γ,F).

In fact, the tangent space T[F ]Wsr,d is identified with the kernel of π>F .

2.1. Prime Fano threefolds of genus 7. We give a brief account ofMukai’s description of a smooth prime Fano threefold of genus 7. Formore details on the material contained in this section, we refer to [Muk88],[Muk89], [Muk95], [IM04a], [Kuz05], [IM07].

We consider thus a smooth prime Fano threefold of genus 7. which wewill denote throughout the paper by X. In particular, X has Picard number1 and the anticanonical class satisfies −KX = HX , where HX is very ampleand its class generates Pic(X). The divisor class HX embeds X in P8 as anACM variety. Remark that Hk,k(X) is generated by the divisor class HX

(for k = 1), the class LX of a line contained in X (for k = 2), the class PX

of a closed point of X (for k = 3). Recall that H2X = 12LX . This allows to

denote the Chern classes c1, c2, c3 of a sheaf F on X by integers.We recall that X is obtained as a smooth linear section of the spinor

tenfold Σ+, sitting in P15. The dual space P15 contains the dual spinortenfold Σ−, and the corresponding orthogonal linear section is a smoothprojective canonical curve Γ of genus 7. The curve Γ can be identifiedwith the moduli space MX(2, 1, 5), and there exists a universal bundle E onX × Γ which makes Γ into a fine moduli space. The curve Γ is called thehomologically projectively dual curve to X.

The spinor varieties Σ± can be seen as the two components of the orthog-onal Grassmann variety of 4-dimensional projective subspaces contained ina smooth quadric in P9. We denote by U± the restrictions to Σ± of thetautological universal subbundle. By a result of Kuznetsov, we have thefollowing natural exact sequences on X × Γ:

0 → E ∗ → U− → G → 0,(2.5)

0 → G → U∗+ → E → 0,(2.6)

where U− and U+ are defined on X ×Γ using pull-backs via the projectionsp : X × Γ → X and q : X × Γ → Γ. Here G is a vector bundle of rank 3and E is the universal bundle mentioned above. Given a point y ∈ Γ (resp.x ∈ X), and a sheaf F on X×Γ, we denote by Fy (resp. Fx) the restrictionof F to X×{y} (resp. to {x}×Γ). The Chern classes of these bundles are:

c1(E ) = HX + HΓ, c2(E ) =712

HX HΓ + 5 LX + η, c3(E ) = 0,


where η ∈ H3(X)⊗H1(Γ) satisfies η2 = 14, and:

c1(U+) = −2HX , c2(U+) = 24LX , c3(U+) = −14PX ,

c1(Gy) = HX , c2(Gy) = 7LX , c3(Gy) = 2PX .

Recall that the vector bundles U+ and Gy, for any y ∈ Γ, are stable by[BF07, Lemma 2.5]. We will also consider the universal exact sequence:

(2.7) 0 → U+ → O10X → U∗+ → 0.

Applying the theorem of Riemann-Roch to a sheaf F on X, of (generic)rank r and with Chern classes c1, c2, c3, we obtain the following formulas:

χ(F ) = r + 3 c1 + 3 c21 −

12c2 + 2 c3

1 −12

c1 c2 +12

c3,

χ(F, F ) = r2 − 12∆(F ).

It is well known that a general hyperplane section S of X is a smooth K3surface (i.e., S has trivial canonical bundle and irregularity zero) of Picardnumber 1 (a generator is the restriction HS of HX to S), and sectional genus7. We recall by [HL97, Part II, Chapter 6] that, given a stable sheaf F ofrank r on S, with Chern classes c1, c2, the dimension at [F ] of the modulispace MS(r, c1, c2) is:

(2.8) ∆(F )− 2 (r2 − 1).

2.2. Derived categories. If Y is a smooth projective variety, we denoteby Db(Y ) its derived category, namely the derived category of complexes ofsheaves on Y with bounded coherent cohomology. We refer to [GM96] and[Wei94] for definitions and notation.

Let now X be a smooth prime Fano threefold of genus 7, Γ the homolog-ically projectively dual curve to X, and E the associated universal bundledefined above. The bundle E is defined on X × Γ, and we denote by pand q the projections of X × Γ to X and Γ. As an essential tool we willuse Kuznetsov’s semiorthogonal decomposition of Db(X), see [Kuz05]. Thistakes the following form:

(2.9) Db(X) ∼= 〈OX ,U∗+,Φ(Db(Γ))〉,

where Φ is the integral functor associated to E defined by:

Φ : Db(Γ) → Db(X), Φ(−) = Rp∗(q∗(−)⊗E ).(2.10)

Recall that the functor Φ is fully faithful, and admits right and left adjointfunctors Φ! and Φ∗ defined by:

Φ! : Db(X) → Db(Γ), Φ!(−) = Rq∗(p∗(−)⊗E ∗(ωΓ))[1],(2.11)

Φ∗ : Db(X) → Db(Γ), Φ∗(−) = Rq∗(p∗(−)⊗E ∗(−HX))[3].(2.12)

The decomposition (2.9) provides a functorial exact triangle:

(2.13) Φ(Φ!(F )) → F → Ψ(Ψ∗(F )),


where Ψ is the inclusion of the subcategory 〈OX ,U∗+〉 in Db(X) and Ψ∗ isthe left adjoint functor to Ψ. The k-th term of the complex Ψ(Ψ∗(F )) canbe written as follows:

(Ψ(Ψ∗(F )))k ∼= Ext−kX (F,OX)∗⊗OX ⊕ Ext1−k

X (F,U+)∗⊗U∗+.

We will also use the following spectral sequences:

Ep,q2 = Extp

X(H−q(a), A) ⇒ Extp+qX (a,A),(2.14)

Ep,q2 = Extp

X(B,Hq(b)) ⇒ Extp+qX (B, b),(2.15)

where a, b are complexes of sheaves on X, and A,B are sheaves on X. Recallthat the maps in these spectral sequences are differentials:

dp,q2 : Ep,q

2 → Ep+2,q−12 .

3. Some vanishing results

In this section, we prove some preliminary vanishing results and we provethat any locally free sheaf in MX(2, 0, 4) is ACM. In all statements, X is asmooth prime Fano threefold of genus 7, S is a general hyperplane sectionsurface of X, C is a general sectional curve of X and F is a locally free sheafin MX(2, 0, 4). We have the following exact sequences, defining respectivelyS and C:

0 → OX(−1) → OX → OS → 0,(3.1)

0 → OS(−1) → OS → OC → 0.(3.2)

Lemma 3.1. The restrictions of U+ and Gy to S are stable vector bundlesfor all y ∈ Γ.

Proof. We will deduce stability from Hoppe’s criterion, see [Hop84, Lemma2.6], see also [AO94, Theorem 1.2]. We have thus to show the followingvanishing results:

H0(S, Gy(−1)) = 0, H0(S,∧2Gy(−1)) = 0,(3.3)

H0(S,U+) = 0, H0(S,U∗+(−1)) = 0(3.4)

H0(S,∧2U+) = 0, H0(S,∧2U∗+(−1)) = 0.(3.5)

Tensoring (3.1) by Gy(−1), we obtain H0(S, Gy(−1)) = 0 since Gy is stableand H1(X, Gy(−2)) = 0, see [BF07, Lemma 2.5]. Note that ∧2Gy

∼= G ∗y (1).

So, tensoring (3.1) by ∧2Gy(−1), we get (3.3), since Gy is stable andH1(X, G ∗

y (−1)) = 0, see again [BF07, Lemma 2.5].Applying the same argument to U+ and U∗+(−1), we get (3.4). Finally, in

view of the proof of [BF07, Lemma 2.5], in order to prove (3.5), it suffices toshow H1(X,∧2U+(−1)) = 0 and H1(X,∧2U∗+(−2)) = 0. This can be checkedvia an easy application of Bott’s theorem on the homogeneous space Σ+. �

Lemma 3.2. For all y ∈ Γ, the restrictions FS and FC of F to the surfaceS and to the curve C satisfy the following conditions:

H0(S, FS) = 0 H0(C,FC) = 0(3.6)

H0(S, FS ⊗ Gy(−t)) = 0 H0(C,FC ⊗ Gy(−t)) = 0 for t ≥ 1(3.7)

H0(S, FS ⊗ U∗+(−t)) = 0 H0(C,FC ⊗ U∗+(−t)) = 0 for t ≥ 1(3.8)


Proof. Let us tensor the exact sequence (3.1) by F . Since H0(X, F ) = 0 bystability and H1(X, F (−1)) ∼= H2(X, F )∗ = 0 by [BF08a, Lemma 4.3], weget the first vanishing in (3.6).

In the proof of Lemma [BF08a, Lemma 4.3] we have also obtainedH1(S, FS(1)) = 0, which implies by Serre duality H1(S, FS(−1)) = 0. Thenby tensoring (3.2) by F , the second vanishing in (3.6) follows.

Recall that Gy(−1) is isomorphic to ∧2G ∗y . Then, dualizing the sequence

(2.5) and restricting to X × {y}, we obtain ∧2G ∗y ↪→ O10

X . Tensoring by FS

we get:H0(S, FS ⊗ ∧2G ∗

y ) ⊆ H0(S, FS)10,

and by (3.6) we conclude that H0(S, FS ⊗ Gy(−1)) = 0. Obviously thisimplies the first vanishing in (3.7) for all t ≥ 1. The second vanishing iseasily obtained replacing FS by FC in the above argument.

In order to prove the third part of the statement, it is enough to provethat the groups H0(S, FS ⊗ Ey(−t)) and H0(C,FC ⊗ Ey(−t)) are both zero.Indeed, in view of (3.7), the relations (3.8) will then easily follow makinguse of the exact sequence (2.6).

Notice that E ∗y∼= Ey(−1), so from the sequence (2.5), restricted to X×{y},

we get Ey(−1) ↪→ O5X . Tensoring by FS , (respectively by FC) and using

(3.6), we obtain H0(S, FS ⊗Ey(−t)) = 0 (respectively H0(C,FC ⊗Ey(−t)) =0) for any t ≥ 1. This completes the proof. �

Lemma 3.3. For all y ∈ Γ we have:

Ext1X(F (1),Gy) = 0.

Proof. Let us first prove that the group Ext1S(FS(1),Gy) ∼= H1(S, FS ⊗Gy(−1)) vanishes. Assume the contrary, and consider the nontrivial ex-tension of the form:

0 → (Gy)S → FS → FS(1) → 0,

where FS is a torsionfree sheaf on S with rank 5 and Chern classes c1(FS) =3, c2(FS) = 47.

Notice now that FS cannot be stable, since the space MS(5, 3, 47) is emptyby the dimension count (2.8). Then the Harder-Narasimhan filtration pro-vides a maximal destabilizing stable quotient Q. Let K be the kernel of theprojection from FS onto Q. Notice that the sheaf K is reflexive by [Har80,Proposition 1.1], since FS if locally free and Q is torsionfree.

Notice that the bundle FS(1) is stable by Maruyama’s theorem, while(Gy)S is stable by Lemma 3.1. Thus, since µ(K) ≥ 3

5 and rk(K) ≤ 4, theonly possible values that the pair (rk(K), c1(K)) can assume are (2, 2) and(3, 2). If the first case takes place, we have that K is a subbundle of F (1)and, since K is reflexive and F (1) is locally free, we have K ∼= F (1). Thismeans that the extension is trivial, a contradiction.

Assume now that rk(K) = 3 and c1(K) = 2. Notice that K has tobe stable since there exist no other possible destabilizing subbundles for F .Hence by the dimension count (2.8) we have c2(K) ≥ 19. On the other handQ is stable with rk(Q) = 2 and c1(Q) = 1, thus by (2.8) we have c2(Q) ≥ 5.


But we have 47 = c2(F ) = c2(K) + c2(Q) + 24 ≥ 48, a contradiction. Thisproves that H1(S, FS ⊗ Gy(−1)) = 0.

Tensoring by FS ⊗Gy(−t) the exact sequence (3.2), and using the secondequality in (3.7), one easily get that H1(S, FS ⊗ Gy(−t)) = 0, for any t ≥ 1.Tensoring now by F ⊗ Gy(−t) the sequence (3.1), by the first vanishing in(3.7), we obtain

H1(X, F ⊗ Gy(−t− 1)) ∼= H1(X, F ⊗ Gy(−t))

for any t ≥ 1. Since this groups vanish for t � 0, we conclude thatExt1X(F (1),Gy) ∼= H1(X, F ⊗ Gy(−1)) = 0. �

Lemma 3.4. For all k 6= 3 we have:

ExtkX(F (1),U+) = 0.

Proof. For k = 0, the statement follows from the stability of F and U+.Applying the functor HomX(F (1),−) to (2.7), we obtain

Ext1X(F (1),U+) ∼= HomX(F (1),U∗+), which vanishes by stability of F

and U∗+, and Ext2X(F (1),U+) ∼= Ext1X(F (1),U∗+). It remains to prove thatthis last group is zero, too.

First we will prove:

(3.9) Ext1S(FS(1),U∗+) ∼= H1(S, FS ⊗ U∗+(−1)) = 0.

Assume by contradiction that there is a nontrivial extension of the form

0 → (U∗+)S → G → FS(1) → 0,

where G is a torsionfree sheaf on S with rank 7 and Chern classes c1(G) = 4,c2(G) = 88. Notice that G cannot be stable, since the space MS(7, 4, 88) isempty by the dimension count (2.8).

Then the Harder-Narasimhan filtration provides a maximal destabilizingstable quotient Q. Let K be the kernel of the projection from G onto Q.Notice that the sheaf K is reflexive by [Har80, Proposition 1.1].

Recall the bundle (U+)S is stable by Lemma 3.1, while FS(1) is stable byMaruyama’s theorem. So, since µ(K) ≥ 4

7 and rk(K) ≤ 6, the only possiblevalues for the pair (rk(K), c1(K)) are (2, 2), (3, 2) and (5, 3).

If the first case takes place, we have that K ∼= F (1), hence the extensionsplits, a contradiction.

Assume now that rk(K) = 3 and c1(K) = 2. Notice that K has to bestable since there exist no other possible destabilizing subbundles for G.Hence by (2.8) we have c2(K) ≥ 19. On the other hand Q is stable withrk(Q) = 4 and c1(Q) = 2, thus by (2.8) we have c2(Q) ≥ 22. But we have88 = c2(G) = c2(K) + c2(Q) + 48 ≥ 89, a contradiction.

Finally assume that rk(K) = 5 and c1(K) = 3. Notice that K has tobe stable since there exist no other possible destabilizing subbundles for G.Hence by (2.8) we have c2(K) ≥ 48. On the other hand Q is stable withrk(Q) = 2 and c1(Q) = 1, thus by (2.8) we have c2(Q) ≥ 5. But we have88 = c2(G) = c2(K) + c2(Q) + 36 ≥ 89, a contradiction. This proves (3.9).

Now, using (3.2) and the second equality in (3.8), one easily getsH1(S, FS ⊗ U∗+(−t)) = 0, for any t ≥ 1. In turn, using (3.1) and the firstvanishing in (3.8), we obtain:

H1(X, F ⊗ U∗+(−t− 1)) ∼= H1(X, F ⊗ U∗+(−t)).


for any t ≥ 1. Since this groups vanishes for t � 0, we conclude thatExt1X(F (1),U∗+) ∼= H1(X, F ⊗ U∗+(−1)) = 0. �

Proposition 3.5. Let X be a smooth prime Fano threefold of genus 7. Thenany locally free sheaf F in MX(2, 0, 4) is ACM.

Proof. We need to prove the following vanishing:

Hk(X, F (t)) = 0,

for all t and for k = 1, 2. Notice that by Serre duality it is enough to proveonly the case k = 1.

Fix a general hyperplane section surface S of X. By the first vanishingin (3.6) and Serre duality, we easily get that H2(S, FS(t)) = 0 for all t ≥0. Now, note that, by [BF08a, Lemma 4.3] and Riemann-Roch, we haveH2(X, F (−1)) = 0. Thus, tensoring (3.1) by F (t), we obtain:

H2(X, F (t)) = 0 for all t ≥ 0,

and by Serre duality it follows H1(X, F (t)) = 0 for all t ≤ −1.We want to prove now that H1(X, F (t)) = 0 for all t ≥ 0. Fix a general

sectional curve C in X and remark that by the second vanishing in (3.6)and by Serre duality we have:

h1(C,FC(t)) = h0(C,FC(−t + 1)) for all t ≥ 1.

Thus, tensoring (3.2) by FS(t) and using the vanishing H1(S, FS) = 0(which holds by Riemann-Roch), we get H1(S, FS(t)) = 0 for any t ≥ 1.Finally, using again the exact sequence (3.1) tensorized by F (t), sinceH1(X, F ) = 0 we get H1(X, F (t)) = 0 for any t ≥ 1, as we wanted. �

Remark 3.6. The previous proposition holds in fact for any smooth primeFano threefold X of genus g ≥ 7. Indeed the same proof works, since [BF08a,Lemma 4.3] can be applied to any locally free sheaf F in MX(2, 0, 4) as soonas mg = d(g + 2)/2e > 4. In turn, this takes place for all g ≥ 7.

4. Proof of the main theorem

This section is devoted to the proof of our main theorem. Let us sketchthe plan of our argument. First of all, by [BF08a, Theorem 4.10] the modulispace MX(2, 0, 4) contains a 5-dimensional reduced irreducible component.Moreover any locally free sheaf in MX(2, 0, 4) is stable by [BF08a, Proposi-tion 4.16] and ACM by Proposition 3.5. Then, Lemma 4.1 will prove that,given a locally free sheaf F in MX(2, 0, 4), the image ϕ(F ) = Φ!(F (1))[−1]is a locally free sheaf on Γ, with rank 2 and degree 4. Then by Corollary(4.3) and Lemma (4.5) we will deduce that in fact ϕ(F ) is contained in theBrill-Noether variety W 1

2,4(Γ). The fact that M`fX (2, 0, 4) is a smooth fivefold

follows by Lemma 4.7. Moreover, it is an open dense subset of W 12,4(Γ) by

Lemma 4.8. Hence the irreducibility of M`fX (2, 0, 4) will follow from that of

W 12,4(Γ), which in turn is proved in [Mer01, Theoreme 4], see also [Mer99].

The result of Mercat holds for any non-hyperelliptic curve, and Γ is so inview of [Muk95, Table 1]. The proof will thus be complete once we establishthe lemmas of this section.


Lemma 4.1. Let X be a smooth prime Fano threefold of genus 7 and F alocally free sheaf in MX(2, 0, 4). Then Φ!(F (1))[−1] is a rank 2 locally freesheaf on Γ, with degree 4.

Proof. Consider the stalk over a point y ∈ Γ of the sheaf Hk(Φ!(F (1))). Wehave:

(4.1) Hk(Φ!(F (1)))y∼= Extk+1

X (Ey, F (1))⊗ωΓ,y.

We would like to prove that this group vanishes for all y ∈ Γ and for allk 6= −1. This amounts to prove that Ext2−k

X (F (1),E ∗y ) = 0 for k = 0, 1, 2.

The case k = 2 follows immediately from the stability of F and Ey.Now let us apply the functor HomX(F (1),−) to the exact sequence (2.5)

restricted to X×{y}. Since HomX(F (1),OX) ∼= Hk(X, F (−1)) = 0 for any k

we have Extk+1X (F (1),E ∗

y ) ∼= Extk(F (1),Gy). Hence in particular the groupExt1X(F (1),E ∗

y ) is zero by the stability of F and Gy (see [BF07, Lemma 2.5]),while the group Ext1X(F (1),E ∗

y ) vanishes by Lemma 3.3.Finally, by Riemann-Roch we have χ(Ey, F (1)) = 2, so the rank of

Φ!(F (1)) is 2. Then we can apply the theorem of Grothendieck-Riemann-Roch to calculate χ(Φ!(F (1))). It easily follows that deg(Φ!(F (1))) = 4. �

Notation. Let F be a sheaf in M`fX (2, 0, 4). We set:

F = Φ!(F (1))[−1].

We set also AF = HomX(U+, F ).

Lemma 4.2. Let F be a sheaf in M`fX (2, 0, 4). Then the following relations

hold:

H0(Φ(F)) ∼= AF ⊗ U∗+,(4.2)

H1(Φ(F)) ∼= F (1),(4.3)

and AF has dimension 2.

Proof. In order to use the decomposition (2.9), we need to com-pute the groups Extk

X(F (1),OX) and ExtkX(F (1),U+). Recall that

ExtkX(F (1),OX) = 0 for all k. On the other hand, by Lemma 3.4 we

know that ExtkX(F (1),U+) = 0 for all k 6= 3. By Riemann-Roch it fol-

lows ext3X(F (1),U+) = 2. Then the exact triangle (2.13) provides thus theisomorphisms (4.2) and (4.3). �

Corollary 4.3. The sheaf F has two independent global sections, andH0(Γ,F) is naturally identified with AF .

Proof. By [Kuz05, Lemma 5.6] we have Φ∗(U∗+) ∼= OΓ and thus:

H0(Γ,F) ∼= HomΓ(OΓ,F) ∼= HomX(U∗+,Φ(F)).

By (4.2) it follows that HomX(U∗+,Φ(F)) ∼= HomX(U∗+,U∗+ ⊗AF ) ∼= AF ,hence we have h0(Γ,F) = 2. �

Lemma 4.4. The vector bundle F is simple.


Proof. We have:

HomΓ(F ,F) ∼= HomX(Φ(F), F (1)[−1]) ∼= HomX(F (1), F (1)).

where the last isomorphism follows immediately by the spectral sequencesince (2.14), setting A = F (1) and a = Φ(F). The claim thus follows fromthe stability of F . �

In fact, the bundle F is not only simple, see the next lemma.

Lemma 4.5. The vector bundle F is stable.

Proof. Assume by contradiction that F is not stable. Then there exists adestabilizing exact sequence on Γ of the form:

(4.4) 0 → L → F →M→ 0,

where L, M are line bundles, ` = deg(L) ≥ 2 and m = deg(M) = 4− `.From (4.2) it follows that for any x ∈ X, h0(Γ,F ⊗ Ex) = 10. Then

tensoring (4.4) by Ex, we have also h0(Γ,L⊗Ex) ≤ 10. From Riemann-Rochit follows that χ(L ⊗ Ex) = 2 ` ≤ 10 and thus ` ≤ 5.

If ` = 5, we have:

H0(Φ(L)) ∼= H0(Φ(F)) ∼= H−1(Φ(Φ!(F (1)))) ∼= U∗ ⊗AF ,

H1(Φ(M)) ∼= H1(Φ(F)) ∼= H0(Φ(Φ!(F (1)))) ∼= F (1),

Hk(Φ(L)) = Hk+1(Φ(M)) = 0, for all k 6= 0.

Therefore, since the functor Φ is fully faithful, we obtain:

Ext1Γ(M,L) ∼= Ext1X(Φ(M),Φ(L)) ∼= Hom(F (1),U∗+ ⊗AF ),

and the last group vanishes by stability of F and U+. This contradictsLemma 4.4.

If 3 ≤ ` ≤ 4, we have h0(Γ,L) ≤ 1 by [Muk95, Table 1]. This easilyimplies h0(Γ,L) = h0(Γ,M) = 1. In particular the line bundle M is eithertrivial either of the form OΓ(y), where y is a point in Γ. Applying the functorΦ to (4.4) and taking cohomology we get a projection from H1(Φ(F)) ∼=F (1) to H1(Φ(M)), hence rk(H1(Φ(M))) ≤ 2. But if M ∼= OΓ we haveH1(Φ(M)) ∼= U+(1) which has rank 5, a contradiction. On the other hand,if M∼= OΓ(y), we can see from the exact sequence:

(4.5) 0 → OΓ →M→ Oy → 0,

that rk(H1(Φ(M))) ≥ 3, again a contradiction.Finally, assume ` = 2. Again we have h0(Γ,L) = h0(Γ,M) = 1, so the

line bundle M is isomorphic to OΓ(Z) where Z is an effective divisor in Γof degree 2. We would like to prove:

(4.6) H1(Φ(M)) ∼= IC(1),

where IC is the ideal sheaf of a conic C ⊂ X, so that, applying Φ to (4.4)we obtain a surjection F (1) → IC(1), and F would be strictly semistable.Recall by [Kuz05, Theorem 5.3] that OZ is isomorphic to Φ!(OC), for someconic C ⊂ X, and thus Φ(OZ) is concentrated in degree zero. Moreover,dualizing the exact sequence (9) in [Kuz05], one gets:

0 → (Φ(OZ))∗(1) → U+(1) → IC(1) → 0.


On the other hand, applying the functor Φ to the exact sequence:

0 → OΓ →M→ OZ → 0,

we obtain an exact sequence:

Φ(OZ) → U+(1) → H1(Φ(M)) → 0.

We get thus (4.6) by the natural isomorphism (Φ(OZ))∗(1) ∼= Φ(OZ), pro-vided by Grothendieck duality, see [BF07, Lemma 2.6], see also [Har66] forgeneral reference. �

Lemma 4.6. Let F be a sheaf in M`fX (2, 0, 4). Then we have:

(4.7) Hk(X,U+⊗F (1)) = 0, for k = 2, 3.

Proof. Recall that in Lemma 4.2 we have proved Hk(X,U∗+⊗F ) = 0 fork 6= 0, so tensoring (2.7) by F we get Hk(X,U+⊗F ) = 0 for k 6= 1. In turn,tensoring (3.1) by U+⊗F (1) and making use of stability of (U+)S and FS

we get H2(S,U+⊗F (1)) = 0. We have thus proved our statement. �

Lemma 4.7. For any pair of sheaves F, F ′ in M`fX (2, 0, 4), we have:

Ext2X(F ′, F ) = 0,

H1(X,U+⊗F (1)) = 0.

Proof. Recall the notation F = Φ!(F (1))[−1] and set F ′ = Φ!(F ′(1))[−1].We have, for all k ∈ Z:

ExtkΓ(F ′,F) ∼= Extk−1

X (Φ(F ′), F (1)),

and by (2.14), we have the spectral sequence:

(4.8) Ep,q2 = Extp

X(H−q(Φ(F ′)), F (1)) ⇒ Extp+qX (Φ(F ′), F (1)).

By Lemma 4.2, we have H0(Φ(F ′)) ∼= AF ′ ⊗ U∗+, and H1(Φ(F ′)) ∼= F ′(1).Using (4.7), the spectral sequence (4.8) becomes:

(4.9) A∗F ′ ⊗HomX(U∗

+, F (1))

d0,02

++XXXXXXXXXXXXXXXXXXXXXXXXXXA∗

F ′ ⊗Ext1X(U∗+, F (1))

d1,02

++XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX0 0

HomX(F ′, F ) Ext1X(F ′, F ) Ext2X(F ′, F ) 0

Since the map d1,02 is zero, we get:

(4.10) A∗F ′ ⊗Ext1X(U∗+, F (1))⊕ Ext2X(F ′, F ) ∼= Ext2Γ(F ′,F).

Note that the group ExtkΓ(F ′,F) vanishes for k ≥ 2 since F and F ′ are

coherent sheaves on a curve. We obtain that both groups H1(X,U+⊗F (1))and Ext2X(F ′, F ) are zero. This proves the lemma. �

Note that the previous Lemma holds even if we take F ′ = F . In particular,for all F in M`f

X (2, 0, 4), we have proved:

Ext2X(F, F ) = 0.

Lemma 4.8. Let F be a sheaf in M`fX (2, 0, 4) and F = Φ!(F (1))[−1]. Then

the two tangent spaces T[F ]W12,4(Γ) and T[F ]MX(2, 0, 4) are naturally identi-

fied.


Proof. Recall first that T[F ]MX(2, 0, 4) is canonically identified withExt1X(F, F ). This space has dimension 5 by Lemma 4.7 and by Riemann-Roch.

On the other hand, the space T[F ]W12,4(Γ) is canonically identified with the

kernel of the transpose π>F of the Petri map, see (2.4). Recall by Corollary4.3 that AF

∼= H0(Γ,F) and consider the natural evaluation map:

ev : AF ⊗OΓ → F ,

and note that the map π>F equals Ext1X(ev,F). By definition of F and sinceΦ! is right adjoint to Φ, this map thus equals:

HomX(Φ(ev), F (1)) : HomX(Φ(F), F (1)) → A∗F ⊗HomX(Φ(OΓ), F (1)).

So this map induces a map of spectral sequences from (4.9) to:

A∗F ⊗HomX(U∗

+, F (1))

++WWWWWWWWWWWWWWWWWWW 0

++WWWWWWWWWWWWWWWWWWWWWWWW 0 0

A∗F ⊗HomX(U+(1), F (1)) 0 0 0

Here, the zeros in the first line are given by Lemma 4.7 and those of thesecond line follow from Lemma 3.4. Thus the kernel of HomX(Φ(ev), F (1))is identified with Ext1X(F, F ). So the two tangent spaces are naturally iden-tified. �

Remark 4.9. The fact that the variety W 12,4(Γ) is irreducible (and nonsin-

gular) relies on its explicit description, obtained in [Mer01, Theoreme 4],and [Mer99, Chapitre 3, Theoreme A.1]. Indeed, an element F of W 1

2,4(Γ)fits into an exact sequence:

0 → O2Γ → F → T → 0,

where T is a torsion sheaf of degree 4 on Γ. Thus the space W 12,4(Γ) is

birational to a P1-bundle over the symmetric power Γ(4).

5. Restricting to a hyperplane section surface

Let again X be a smooth prime Fano threefold of genus 7, and consider therestriction FS of a sheaf F in the moduli space M`f

X (2, 0, 4) to a hyperplanesection surface S of X. For general S, the sheaf FS thus belongs to the mod-uli space M`f

S (2, 0, 4). Recall by [Muk84] that the moduli space M`fS (2, 0, 4)

is a symplectic manifold. Following an idea of Tyurin, we prove here thatthe restriction mapping is injective, hence that M`f

X (2, 0, 4) is lagrangian inM`f

S (2, 0, 4).

Proposition 5.1. Let S be a smooth hyperplane section surface of X withPic(S) ∼= 〈HS〉. Then the restriction map:

ρ : M`fX (2, 0, 4) → M`f

S (2, 0, 4)F 7→ FS

is a closed embedding, and Im(ρ) is a lagrangian submanifold of M`fS (2, 0, 4).


Proof. The image of the restriction map ρ is a lagrangian submanifold by[Tyu04]. Thus we need only prove that ρ is well-defined and injective every-where.

Let thus F be a sheaf in M`fX (2, 0, 4) and FS be its restriction to S. First

note that the sheaf FS is a stable vector bundle. Indeed the first vanishingin (3.6) takes place for any hyperplane section surface S, and this impliesstability by Hoppe’s criterion since Pic(S) ∼= 〈HS〉. Therefore ρ is well-defined.

In order to prove that ρ is injective, we let F ′ be a sheaf in M`fX (2, 0, 4),

not isomorphic to F and we set F ′S for its restriction to S. Let us see that

the existence of an isomorphism α : FS → F ′S leads to a contradiction.

Tensoring (3.1) with F ′ provides a surjective map F → F ′S . We want to

prove that this map lifts to a map α : F → F ′, and we note that this is thecase if the obstruction group Ext1X(F, F ′(−1)) vanishes. But this group isdual to Ext1X(F ′, F ), which vanishes by Lemma 4.7. Therefore we have themap α, and, by stability of F and F ′, the map α must be an isomorphism.This is a contradiction. �

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E-mail address: [email protected], [email protected]

Dipartimento di Matematica “G. Castelnuovo”, Universita di RomaSapienza, Piazzale Aldo Moro 5, I-00185 Roma - Italia

URL: http://web.math.unifi.it/users/brambill/

E-mail address: [email protected]

Universite de Pau et des Pays de l’Adour, Av. de l’Universite - BP 576 -64012 PAU Cedex - France

URL: http://web.math.unifi.it/users/faenzi/

http://web.math.unifi.it/users/brambill/

http://web.math.unifi.it/users/faenzi/

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